Covering with Hecke points
Let \({\mathcal H}^n\) be the \(GL_n (\mathbb{R})\) upper-half-space \({\mathcal H}^n = GL_n (\mathbb{R})/O(n, \mathbb{R})Z\), where \(O(n, \mathbb{R})\) is the orthogonal group and \(Z\) the subgroup of scalar matrices. Consider the quotient space \({\mathcal M}_n = SL(n, \mathbb{Z}) \backslash {\mathcal H}^n\). One may endow this space with a natural metric. The author studies the distribution of Hecke points in \({\mathcal M}_3\). For \(p\) a prime, the \(p\)-Hecke points are by definition the cosets represented by the upper triangular integral matrices of determinant \(p\). Focusing on the case \(n = 3\), he shows that given a bounded set \(\Omega\) of \({\mathcal M}_3\) there exists a number \(N_\Omega\) such that for all primes \(p > N_\Omega\) any point of \(\Omega\) is inside a ball of radius \(Kp^{- 1/10}\) centered at some \(p\)-Hecke point. Here \(K\) is an absolute constant. In particular, this implies that the Hecke points are dense. The exponent \(- 1/10\) is based on the estimate for the Hecke eigenvalues of the \(GL_3\) cusp forms due to \textit{H. Jacquet}, \textit{I. I. Piatetski-Shapiro}, and \textit{J. Shalika} [Ann. Math., II. Ser. 109, 169-212 (1979; Zbl 0401.10037)]; the Ramanujan conjecture would give exponent \(- 1/5\). The author remarks that the same proof works when \(n = 2\), and, using the bound of \textit{D. Bump}, \textit{W. Duke}, \textit{J. Hoffstein} and \textit{H. Iwaniec} [Int. Math. Res. Not. 1992, No. 4, 75-81 (1992; Zbl 0760.11017)], gives a similar result with radius \(Kp^{-9/56}\). The proof is based on harmonic analysis on \({\mathcal M}_3\). For \(\varepsilon > 0\), the author shows that there is a point-pair invariant \(K_\varepsilon (x,y)\), corresponding to a positive definite integral operator, which is nonzero only when the distance \(d(x,y) < \varepsilon\). He then applies the Hecke operator \(T_p\) to this function (in \(y)\), and analyzes the result by means of the \(GL_3\) Spectral Theorem to obtain his theorem. The author also applies this method on the group level rather than the symmetric space level, focusing this time on \(GL_2(\mathbb{R})\). Doing so, he proves that given a matrix \(A = (a_{ij}) \in SL_2(\mathbb{R})\) and a prime number \(p\), there exists an integral matrix \(B = (b_{ij})\) of determinant \(p\) such that \(\max_{i,j} |p^{- 1/2} b_{ij} - a_{ij} |< Cp^{- 3/28}\), where \(C\) is a constant depending only on \(\max_{i,j} |a_{ij} |\). Though the exponent \(- 3/28\) is not as strong as \textit{G. Harman}'s result of \(- 1/8\) [J. Number Theory 34, No. 1, 63-81 (1990; Zbl 0699.10053)], progress on the Ramanujan conjecture would improve the result. In particular, the Ramanujan conjecture itself implies exponent \(- 1/6\). Interestingly, this is the same bound which could be obtained from Harman's method if one knew Hooley's \(R^*\) conjecture.
scientific article; zbMATH DE number 792467